Frege – Page 2 – Maverick Philosopher

Kenny, Geach, and the Perils of Reading Frege Back Into Aquinas

London Ed has informed me of the passing of Peter Geach. May he find the Unchanging Light that he sought through his long and productive life of truth-seeking in these shadowlands. One honors a thinker best by thinking his thoughts, sympathetically, but critically. Here is one of my attempts. Others referenced below.

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I have been studying Anthony Kenny, Aquinas on Being (Oxford 2002). I cannot report that I find it particularly illuminating. I am troubled by the reading back of Fregean doctrines into Aquinas, in particular in the appendix, "Frege and Aquinas on Existence and Number." (pp. 195-204) Since Kenny borrows heavily from Peter Geach, I will explain one of my misgivings in connection with a passage from Geach's important article, "Form and Existence" in God and the Soul. Geach writes,

Frege, like Aquinas, held that there was a fundamental distinction in rebus answering to the logical distinction between subject and predicate — the distinction between Gegenstand (object) and Begriff (concept). [. . .] And for Frege the Begriff, and it alone, admits of repetition and manyness; an object cannot be repeated — kommt nie wiederholdt vor. (45-46)

So far, so good. Geach continues:

Understood in this way, the distinction between individual and form is absolutely sharp and rigid; what can be sensibly said of one becomes nonsense if we try to say it of the other. [. . .] Just because of this sharp distinction, we must reject the Platonic doctrine that what a predicate stands for is is some single entity over against its many instances, hen epi pollon. On the contrary: the common nature that the predicate 'man' (say) stands for can be indifferently one or many, and neither oneness nor manyness is a mark or note of human nature itself. This point is made very clearly by Aquinas in De Ente et Essentia. Again we find Frege echoing Aquinas; Frege counts oneness or manyness (as the case may be) among the properties (Eigenschaften) of a concept, which means that it cannot at the same time be one of the marks or notes (Merkmalen) of that concept. (46)

I smell deep confusion here. But precisely because the confusion runs deep I will have a hard time explaining clearly wherein the confusion consists. I will begin by making a list of what Geach gets right.

1. Objects and individuals are unrepeatable.
2. Concepts and forms are repeatable.
3. Setting aside the special question of subsistent forms, no individual is a form, and no object is a concept.
4. Frege distinguishes between the marks of a concept and the properties of a concept. The concept man, for example, has the concept animal as one of its marks. But animal is not a property of man, and this for the simple reason that no concept is an animal. Man has the property of being instantiated. This property, however, is not a mark of man since it is not included within the latter's conceptual content: one cannot by sheer analysis of the concept man determine whether or not there are any men. So there is a sense in which "neither oneness nor manyness is a mark or note of human nature itself." This is true if taken in the following sense: neither being instantiated singly nor being instantiated multiply is a mark of the concept man.

But how do these points, taken singly or together, support Geach's rejection of "the Platonic doctrine that what the predicate stands for is some single entity over against its many instances"? They don't!

It seems obvious to me that Geach is confusing oneness/manyness as the relational property of single/multiple instantiation with oneness/manyness as the monadic property of being one or many. It is one thing to ask whether a concept is singly or multiply instantiated. It is quite another to ask whether the concept itself is one or many. It is also important to realize that a Fregean first-level concept, when instantiated, does not enter into the structure of the individuals that instantiate it. Aquinas is a constituent ontologist, but Frege is not. This difference is deep and causes a world of trouble for those who attempt to understand Aquinas in Fregean terms. For Frege, concepts are functions, and no function enters into the structure of its argument. The propositional function x is a man is not a constituent of Socrates. What's more, the value of the function for Socrates as argument is not a state of affairs with Socrates and the function as constituents. The value of the function for Socrates as argument is True; for Stromboli as argument, False. And now you know why philosophers speak of truth-values. It's mathematical jargon via Frege the mathematician.

The Fregean concept man is one, not many. It is one concept, not many concepts. Nor is it neither one nor many. It can have one instance, or many instances, or no instance. The Thomistic form man, however, is, considered in itself, neither one nor many. It is one in the intellect but (possibly) many in things. In itself, however, it is neither. And so it is true to say that the form is not "some single entity over against its many instances." It is not a single entity because, considered in itself, it is neither single nor multiple.

But this doesn't follow from point (3) above. And therein consists Geach's mistake. One cannot validly move from the "sharp distinction" between individuals/objects and forms/concepts to the conclusion that what a predicate stands for is not a single entity. Geach makes this mistake because of the confusion exposed two paragraphs supra. The mutual exclusion of objects and concepts does not entail that concepts cannot be single entities.

There is another huge problem with reading Frege back into Aquinas, and that concerns modes of existence (esse). A form in the intellect exists in a different way than it does in things. But if Frege is right about existence, there cannot be modes of existence. For if existence is instantiation, then there cannot be modes of existence for the simple reason that there cannot be any modes of instantiation.

I'll say more about this blunder in another post. It rests in turn on a failure to appreciate the radically different styles of ontology practiced by Aquinas and Frege. In my jargon, Aquinas is a constituent ontologist while Frege is a nonconstituent ontologist. In the jargon of Gustav Bergmann, Aquinas is a compex ontologist while Frege is a function ontologist.

Stanislav Sousedik’s “Towards a Thomistic Theory of Predication”

Enough of politics, back to some hard-core technical philosophy. If nothing else, the latter offers exquisite escapist pleasures not unlike those of chess. Of course I don't believe that technical philosophy is escapist; my point is a conditional one: if it is, its pleasures suffice to justify it as a form of recuperation from this all-too-oppressive world of 'reality.' It's what I call a 'fall-back position.'

I have been commissioned to review the collection of which the above-captioned article is a part. The collection is entitled Metaphysics: Aristotelian, Scholastic, Analytic (Ontos Verlag 2012) and includes contributions by Peter van Inwagen, Michael Loux, E. J. Lowe, and several others. My review article will address such topics as predication, truth-makers, bare particulars, and the advantages and liabilities of constituent ontology. I plan a series of posts in which I dig deep into some of the articles in this impressive collection.

Stanislav Sousedik argues for an "identity theory of predication" according to which a predicative sentence such as 'Peter is a man' expresses an identity of some sort between the referent of the subject 'Peter' and the referent of the predicate 'man.' Now to someone schooled in modern predicate logic (MPL) such an identity theory will appear wrongheaded from the outset. For we learned at Uncle Gottlob's knee to distinguish between the 'is' of identity ('Peter is Peter') and the 'is' of predication ('Peter is a man').

But let's give the Thomist theory a chance. Sousedik, who is well aware of Frege's distinction, presents an argument for the identity in some sense of subject and predicate. He begins by making the point that in the declarative 'Peter is a man' and the vocative 'Peter, come here!' the individual spoken about is (or can be) the same as the individual addressed. But common terms such as 'man' can also be used to address a person. Instead of saying, 'Peter, come here!' one can say 'Man, come here!' And so we get an argument that I will put as follows:

1. Both 'Peter' and 'man' can be used to refer to the same individual. Therefore

2. A common term can be used to refer to an individual. But

3. Common terms also refer to traits of individuals. Therefore

4. The traits must be identical in some sense to the individuals. E.g., the referent of 'Peter' must be in some sense identical to the referent of 'man.'

But in what sense are they identical? Where Frege distinguishes between predication and identity, Sousedik distinguishes between weak and strong identity. 'Peter is Peter' expresses strong identity while 'Peter is a man' expresses weak identity. "Strong identity is reflexive, symmetric, and transitive, weak identity has none of these formal properties." (254) It thus appears that strong identity is the same as what modern analytic philosophers call (numerical) identity. It is clear that 'Peter is a man' cannot be taken to express strong identity. But what is weak identity?

S. is a constituent ontologist. He holds that ordinary substances such as Peter have what he calls "metaphysical parts." Whereas Peter's left leg is a physical part of him, his traits are metaphysical parts of him. Thus the referents of the common terms 'man,' 'animal,' living thing,' etc. are all metaphysical parts of Peter. Clearly, these are different traits of Peter. But are they really distinct in Peter? S. says that they are not: they are really identical in Peter and only "virtually distinct" in him. The phrase is defined as follows.

(Def. 1) Between x, y there is a virtual distinction iff (i) x, y are really identical; (ii) x can become an object of some cognitive act Φ without y being the object of the same act Φ . . . . (251)

For example, humanity and animality in Peter are really identical but virtually distinct in that humanity can be the intentional object of a cognitive act without animality being the object of the same act. I can focus my mental glance so to speak on Peter's humanity while leaving out of consideration his animality even though he is essentially both a man and an animal and even though animality is included within humanity.

The idea, then, is that Peter has metaphysical parts (MPs) and that these items are really identical in Peter but virtually distinct, where the virtual distinctness of any two MPs is tied to the possibility of one of them being the object of a cognitive act without the other being the object of the same act.

Is S. suggesting that virtual distinctness is wholly mind generated? I don't think so. For he speaks of a potential distinction of MPs in concrete reality, a distinction that becomes actual when the understanding grasps them as distinct. (253) And so I take the possibility mentioned in clause (ii) of the above definition to be grounded not only in the mind's power to objectify and abstract but also in a real potentiality in the MPs in substances like Peter.

One might be tempted to think of weak identity as a part-whole relation. Thus one might be tempted to say that 'Peter' refers to Peter and 'man' to a property taken in the abstract that is predicable not only of Peter but of other human beings as well. 'Peter is a man' would then say that this abstract property is a metaphysical part of Peter. But this is not Sousedik's or any Thomist's view. For S. is committed to the idea that "Every empirical individual and every part or trait of it is particular." (251) It follows that no metaphysical part of any concrete individual is a universal. Hence no MP is an abstract property. So weak identity is not a part-whole relation.

What is it then?

First of all, weak identity is a relation that connects a concrete individual such as Peter to a property taken abstractly. But in what sense is Peter identical to humanity taken abstractly? In this sense: the humanity-in-Peter and the humanity-in-the-mind have a common constituent, namely, humanity taken absolutely as common nature or natura absoluta or natura secundum se. (254) What makes weak identity identity is the common constituent shared by the really existing humanity in Peter and the intentionally existing humanity in the mind of a person who judges that Peter is human.

So if we ask in what sense the referent of 'Peter' is identical to the referent of 'man,' the answer is that they are identical in virtue of the fact that Peter has a proper metaphysical part that shares a constituent with the objective concept referred to by 'man.' Sousedik calls this common constituent the "absolute subject." In our example, it is human nature taken absolutely in abstraction from its real existence in Peter and from its merely intentional existence in the mind.

Critical Observations

I am deeply sympathetic to Sousedik's constituent-ontological approach, his view that existence is a first-level 'property,' and the related view that there are modes of existence. (253) But one of the difficulties I have with S.'s identity theory of predication is that it relies on common natures, and I find it difficult to make sense of them as I already spelled out in a previous post. Common natures are neither one nor many, neither universal nor particular. Humanity is many in things but one in the mind. Hence taken absolutely it is neither one nor many. It is this absolute feature that allows it be the common constituent in humanity-in-Peter and humanity-in-the-mind. And as we just saw, without this common constituent there can be no talk of an identity between Peter and humanity. The (weak) identity 'rides on' the common constituent, the natura absoluta. Likewise, humanity is particular in particular human beings but universal in the mind (and only in the mind). Hence taken absolutely it is neither particular nor universal.

But it also follows that the common nature is, in itself and taken absolutely, neither really existent nor intentionally existent. It enjoys neither esse naturale (esse reale) nor esse intentionale. Consequently it has no being (existence) at all. This is not to say that it is nonexistent. It is to say that it is jenseits von Sein und Nichtsein to borrow a phrase from Alexius von Meinong, "beyond being and nonbeing."

The difficulty is to understand how there could be a plurality of distinct items that are neither universal nor particular, neither one nor many, neither existent nor nonexistent. Note that there has to be a plurality of them: humanity taken absolutely is distinct from animality taken absolutely, etc. And what is the nature of this distinctness? It cannot be mind-generated. This is because common natures are logically and ontologically prior to mind and matter as that which mediates between them. They are not virtually distinct. Are they then really distinct? That can't be right either since they lack esse reale.

And how can these common or absolute natures fail to be, each of them, one, as opposed to neither one nor many? The theory posits a plurality of items distinct among themselves. But if each is an item, then each is one. An item that is neither one nor many is no item at all.

There is also this consideration. Why are common natures more acceptable than really existent universals as constituents of ordinary particulars such as Peter? The Thomists allow universals only if they have merely intentional existence, existence 'in' or rather for a mind. "Intentional existence belongs to entities which exist only in dependence upon the fact that they are objects of our understanding." (253) They insist that, as S. puts it, "Every empirical individual and every part or trait of it is particular." (251) S. calls the latter an observation, but it is not really a datum, but a bit of theory. It is a datum that 'man' is predicable of many different individuals. And it is a datum that Peter is the subject of plenty of essential predicates other than 'man.' But it is not a clear datum that Peter is particular 'all the way through.' That smacks of a theory or a proto-theory, not that it is not eminently reasonable.

One might 'assay' (to use G. Bergmann's term) an ordinary particular as a complex consisting of a thin or 'bare' particular instantiating universals. This has its own difficulties, of course, but why should a theory that posits common natures be preferrable to one that doesn't but posits really existent universals instead? Either way problems will arise.

The main problem in a nutshell is that it is incoherent to maintain that some items are such that they have no being whatsoever. 'Some items are such that they have no being whatsoever' is not a formal-logical contradiction, pace van Inwagen, but it is incoherent nonetheless. Or so it seems to me.

Aquinas Meets Frege: Analysis of an Argument from De Ente et Essentia

The other day I expressed my reservations as to the coherence of the Thomistic notion of a common nature. Let's plunge a little deeper by considering the argument from Chapter 3 of Thomas Aquinas, On Being and Essence (tr. Robert T. Miller, emphasis added):

The nature, however, or the essence thus understood can be considered in two ways.
First, we can consider it according to its proper notion, and this is to consider it absolutely. In this way, nothing is true of the essence except what pertains to it absolutely: thus everything else that may be attributed to it will be attributed falsely. For example, to man, in that which he is a man, pertains animal and rational and the other things that fall in his definition; white or black or whatever else of this kind that is not in the notion of humanity does not pertain to man in that which he is a man. Hence, if it is asked whether this nature, considered in this way, can be said to be one or many, we should concede neither alternative, for both are beyond the concept of humanity, and either may befall the conception of man. If plurality were in the concept of this nature, it could never be one, but nevertheless it is one as it exists in Socrates. Similarly, if unity were in the notion of this nature, then it would be one and the same in Socrates and Plato, and it could not be made many in the many individuals. Second, we can also consider the existence the essence has in this thing or in that: in this way something can be predicated of the essence accidentally by reason of what the essence is in, as when we say that man is white because Socrates is white,
although this does not pertain to man in that which he is a man.

The argument may be set forth as follows:

1. A nature can be considered absolutely or according to the being it has in this or that individual.

2. If a nature is considered absolutely, then it is not one. For if oneness were included in the nature of humanity, e.g., then humanity could not exist in many human beings.

3. If a nature is considered absolutely, then it is not many. For if manyness were included in the nature of humanity, e.g., then humanity could not exist in one man, say, Socrates.

Therefore

4. If a nature is considered absolutely, then it is neither one nor many, neither singular nor plural.

I find this argument intriguing because I find it extremely hard to evaluate, and because I find the conclusion to be highly counterintuitive. It seems to me obvious that a nature or essence such as humanity is one, not many, and therefore not neither one nor many!

The following is clear. There are many instances of humanity, many human beings. Therefore, there can be many such instances. It follows that there is nothing in the nature of humanity to preclude there being many such instances. But there is also nothing in the nature of humanity to require that there be many instances of humanity, or even one instance. We can express this by saying that the nature humanity neither requires nor precludes its being instantiated. It allows but does not entail instantiation. This nature, considered absolutely, logically allows multiple instantiation, single instantiation, and no instantiation. It logically allows that there be many men, just one man, or no men.

That much is crystal clear. But surely it does not follow that the nature humanity is neither one nor
many. What Aquinas is doing above is confusing what Frege calls a mark (Merkmal) of a concept with a property (Eigenschaft) of a concept. (See Foundations of Arithmetic, sec. 53, first publ. 1884) The marks of a concept are the subconcepts which are included within it. Thus man has animal and rational as marks. But these are not properties (Eigenschaften) of the concept man since no concept is an animal or is rational. Being instantiated is an example of a property of man, a property that cannot be a mark of man. If being instantiated were a mark of man, then the concept man could not fail to be instantiated. In general, the marks of a concept are not properties thereof, and vice versa.

A couple more examples. Three-sided is a mark of the concept triangle, but is is not a property of this concept for the simple reason that no concept is three-sided. Male is a mark of the concept bachelor, but not a property of it since no concept is male.

Aquinas has an insight which can be expressed in Fregean jargon as follows. Being singly instantiated — one in reality – and being multiply instantiated — many in reality — are not marks (Merkmale) of the nature humanity. But because he (along with everyone else prior to 1884) confuses marks with properties (Eigenschaften), he concludes that the nature itself cannot be either one or many. But surely the nature itself is one, nor many. That is consistent with holding that the nature admits of single instantiation, multiple instantiation, or no instantiation.

To put it another way, Aquinas confuses the 'is' of predication ('Socrates is a man') with the 'is' of subordination ('Man is an animal'). Man is predicable of Socrates, but animal is not predicable of man, pace Aristotle, Categories 3b5: no concept or nature is an animal. Socrates falls under man; Animal falls within man. Falling-under and falling-within are different relations. Animal is superordinate to man while man is subordinate to animal. But that is not to say that animal is predicable of man. Both animal and man are predicable of Socrates, which is to say: Socrates falls under both concepts. But man does not fall under animal, animal falls within man. If man fell under animal, then the concept man would be an animal, which is absurd.

For these reasons I do not find the argument from De Ente et Essentia compelling. It is based on confusions that the great logician Gottlob Frege was the first to sort out. But perhaps there is a good Thomist response.

Still Trying to Understand Van Inwagen’s Half-Way Fregeanism about Existence

In section 53 of The Foundations of Arithmetic, Gottlob Frege famously maintains that

. . . existence is analogous to number. Affirmation of existence is in fact nothing but denial of the number nought. Because existence is a property of concepts the ontological argument for the existence of God breaks down. (65)

Frege is here advancing a double-barreled thesis that splits into two subtheses.

ST1. Existence is analogous to number.

ST2. Existence is a property (Eigenschaft) of concepts and not of objects.

In the background is the sharp distinction between property (Eigenschaft) and mark (Merkmal). Three-sided is a mark of the concept triangle, but not a property of this concept; being instantiated is a property of this concept but not a mark of it. The Cartesian-Kantian ontological argument "from mere concepts" (aus lauter Begriffen), according to Frege, runs aground because existence cannot be a mark of any concept, but only a property of some concepts. And so one cannot validly argue from the concept of God to the existence of God.

Existence as a property of concepts is the property of being-instantiated. We can therefore call the Fregean account of existence an instantiation account.

My concern in this entry is the logical relation between the subtheses. Does the first entail the second or are they logically independent? There is a clear sense in which (ST1) is true. Necessarily, if horses exist, then the number of horses is not zero, and vice versa. 'So 'Horses exist' is logically equivalent to 'The number of horses is not zero.' This is wholly unproblematic for those of us who agree that there are no Meinongian nonexistent objects. But note that, in general, equivalences, even logical equivalences, do not sanction reductions or identifications. So it remains an open question whether one can take the further step of reducing existence to instantiation, or identifying existence with instantiation, or even eliminating existence in favor of instantiation.

(ST1), then, is unproblematically true if understood as expressing the following logical equivalence: 'Necessarily Fs exist iff the number of Fs is not zero.' My question is whether (ST1) entails (ST2). Peter van Inwagen in effect denies the entailment by denying that the 'the number of . . . is not zero' is a predicate of concepts:

I would say that, on a given occasion of its use, it predicates of certain things that they number more than zero. Thus, if one says, 'The number of horses is not zero,' one predicates of horses that they number more than zero. 'The number of . . . is not zero' is thus what some philosophers have called a 'variably polyadic' predicate. But so are many predicates that can hardly be regarded as predicates of concepts. The predicates 'are ungulates' and 'have an interesting evolutionary history,' for example, are variably polyadic predicates. When one says, 'Horses are ungulates' or 'Horses have an interesting evolutionary history' one is obviously making a statement about horses and not about the concept horse. ("Being, Existence, and Ontological Commitment," pp. 483-484)

It is this passage that I am having a hard time understanding. It is of course clear what van Inwagen is trying to show, namely, that the Fregean subtheses are logically independent and that one can affirm the first without being committed to the second. One can hold that existence is denial of the number zero without holding that existence is a property of concepts.

But I am having trouble with the claim that the predicate 'the number of . . . is not zero' is 'variably polyadic' and the examples van Inwagen employs. 'Robbed a bank together' is an example of a variably polyadic predicate. It is polyadic because it expresses a relation and it is variably polyadic because it expresses a family of relations having different numbers of arguments. For example, Bonnie and Clyde robbed a bank together, but so did Ma Barker and her two boys, Patti Hearst and three members of the ill-starred Symbionese Liberation Army, and so on. (Example from Chris Swoyer and Francesco Orilia.)

Now when I say that the number of horses is not zero, what am I talking about? It is plausible to say that I am talking about horses, not about the concept horse. What I don't understand is why van Inwagen says that 'the number of . . . is not zero' is a variably polyadic predicate. As far as I can see, it is not even polyadic, let alone variably polyadic. What is the relation that the predicate expresses, and why is that relation multigrade? I grant that there are indefinitely many ways the number of horses could be not zero: there could be one horse, two, three, and so on. But what is the relation between or among horses that this supposedly polyadic predicate expresses?

'. . .exist(s)' is monadic. It expresses no relation. Why not say the same about 'such that their number is not zero'?

Now consider 'are ungulates.' If an ungulate is just a mammal with hooves, then I fail to see how 'are ungulates' is polyadic, let alone variably polyadic. 'Are hooved mammals' is monadic.

The other example is 'Horses have an interesting evolutionary history.' This sentence is clearly not about the concept horse. But it is not about any individual horse either. Consider Harry the horse. Harry has a history. He was born in a certain place, grew up, was bought and sold, etc. and then died at a certain age. He went through all sorts of changes. But Harry didn't evolve, and so he had no evolutionary history. No individual evolves; populations evolve:

Evolutionary change is based on changes in the genetic makeup of populations over time. Populations, not individual organisms, evolve. Changes in an individual over the course of its lifetime may be developmental (e.g., a male bird growing more colorful plumage as it reaches sexual maturity) or may be caused by how the environment affects an organism (e.g., a bird losing feathers because it is infected with many parasites); however, these shifts are not caused by changes in its genes.
While it would be handy if there were a way for environmental changes to cause
adaptive changes in our genes — who wouldn't want a gene for malaria resistance
to come along with a vacation to Mozambique? — evolution just doesn't work that
way. New gene variants (i.e., alleles) are produced by random mutation, and over the course of many generations, natural selection may favor advantageous variants, causing them to become more common in the population.

'Horses have an interesting evolutionary history,' then, is not about the concept horse or about any individual horse. The predicate in this sentence appears to be non-distributive or collective. It is like the predicate in 'Horses have been domesticated for millenia.' That is certainly not about the concept horse. No concept can be ridden or made to carry a load. But it is also not about any individual horse. Not even the Methuselah of horses, whoever he might be, has been around for millenia.

A predicate F is distributive just in case it is analytic that whenever some things are F, then each is F. Thus a distributive predicate is one the very meaning of which dictates that if it applies to some things, then it applies to each of them. 'Blue' is an example. If some things are blue, then each of them is blue.

If a predicate is not distributive, then it is non-distributive (collective). If some Occupy-X nimrods have the building surrounded, it does not follow that each such nimrod has the building surrounded. If some students moved a grand piano into my living room, it does not follow that each student did. If bald eagles are becoming extinct, it does not follow that each bald eagle is becoming extinct. Individual animals die, but no individual animal ever becomes extinct. If the students come from many different countries, it does not follow that each comes from many different countries. If horses have an interesting evolutionary history, it does not follow that each horse has an interesting evolutionary history.

My problem is that I don't understand why van Inwagen gives the 'Horses have an interesting evlutionary history' example when he is committed to saying that each horse exists. His view , I take it, is that 'exist(s)' is a first-level non-distributive predicate. 'Has an interesting evolutionary history,' however, is a first-level non-distributive predicate. Or is it PvI's view that 'exist(s)' is a first-level non-distributive predicate?

Either I don't understand van Inwagen's position due to some defect in me, or it is incoherent. I incline toward the latter. He is trying to show that (ST1) doe not entail (ST2). He does this by giving examples of predicates that are first-level, i.e., apply to objects, but are variably polyadic as he claims 'the number of . . . is not zero' is variably polyadic. But the only clear example he gives is a predicate that is non-distributive, namely 'has an interesting evilutionary history.' 'Horses exist,' however, cannot be non-distributive. If some horses exist, then each of them exists. And if each of them exists, then 'exists' is monadic, not polyadic, let alone variably polyadic.

The ComBox is open if there is anyone who knows this subject and has read PvI's paper and can set me straight.

Existence and Plural Predication: Could ‘Exist(s)’ be a First-Level Non-Distributive Predicate?

'Horses exist' is an example of an affirmative general existential sentence. What is the status of the predicate '___ exist' in such a sentence? One might maintain that 'exist(s)' is a second-level predicate; one might maintain that it is a first-level distributive predicate; one might maintain that it is a first-level non-distributive (collective) predicate.

1. Frege famously maintained that 'exist(s)' is a second-level predicate, a predicate of concepts only, and never a first-level predicate, a predicate of objects. Russell followed him in this. A consequence of this view is that 'Horses exist' is not about what it seems to be about, and does not say what it seems to say. It seems to be about horses, and seems to say of them that they exist. But on Frege's analysis the sentence is about the concept horse and says of it, not that it exists, but that it has one or more instances.

Paradoxically, the sentence ''Horses exist' on Frege's analysis says about a non-horse something that cannot be true of a horse or of any concrete thing!

For an interesting comparison, consider 'Horses surround my house.' Since no horse could surround my house, it is clear that the sentence is not about each of the horses that surround my house. What then is it about? One will be tempted to reach for some such singularist analysis as: 'A set of horses surrounds my house.' But this won't do since no such abstract object as a set could surround anything. So if the sentence is really about a set of horses then it cannot say what it appears to say. It must be taken to say something different from what it appears to say. So what does 'Horses surround my house' say about a set if it is about a set?

One might be tempted to offer this translation: 'A set of horses is such that its members are surrounding my house.' But this moves us in a circle, presupposing as it does that we already understand the irreducibly plural predication 'Horses surround my house.' After all, if the members of a set of horses surround my house that is no different from horses surrounding my house.

The circularity here is structurally similar to that of the Fregean analysis. If 'Horses exist' is about a concept, and says of that concept that it has instances, then of course those instances are horses that exist. So the attempt to remove existence from individuals and make of it a property of concepts ends up reinstating existence as a 'property' of individuals.

Pursuing the analogy a bit further, the refusal to grant that there are irreducibly plural predications such as 'Horses surround my house' is like the refusal to grant that there are irreducibly first-level existence sentences.

2. Pursuing the analogy still further, is it possible to construe the predicate in 'Horses exist' as a non-distributive first-level predicate like the predicate in 'Horses surround my house'? First some definitions.

I will assume for the purposes of this post that 'Horses surround my house' and 'Horses have an interesting evolutionary history' are irreducibly plural predications. (That they are plural is obvious; that they are irreducibly plural is not. For arguments see Thomas McKay, Plural Predication.) And of course they are first-level as well: they are about horses, not about concepts or properties or propositional functions. Now is 'Horses exist' assimilable to 'Horses surround my house' or is it assimilable to 'Horses are four-legged'? The predicate in the later is a distributive first-level predicate, whereas the predicate in 'Horses surround my house' is a non-distributive first-level predicate.

I am assuming that the 'Fressellian' second-level analysis is out, but I won't repeat the arguments I have given ad nauseam elsewhere.

I do not understand how 'exist(s)' could be construed as a non-distributive predicate. For if it is non-distributive, then it is possible that some things exist without it being the case that each of them exists. And that I do not understand. If horses exist, then each horse exsts.

Peter van Inwagen seems (though it not clear to me) to be saying that 'exists(s)' is a non-distributive first-level predicate. He compares 'Horses exist' to 'Horses have an interesting evolutionary history.' 'Horses exist,' he tells us, is equivalent to 'The number of horses is not zero.' ("Being, Existence, and Ontological Commitment," p. 483) But he denies that 'exists(s)' is second-level. To say that the number of horses is not zero is to predicate of horses that they number more than zero. (483) It is not to predicate of the concept horse that the cardinality of its extension is more than zero.

Now we cannot say of a horse that it surrounds a house or has an interesting evolutionary history. We can say that of horses, but not of a horse. Can we say of a horse that it numbers more than zero? We can of course say of horses that they number more than zero. But I don't see how we can sensibly say of an individual horse that it numbers more than zero. Perhaps Frege was wrong to think that number words can only be predicates of concepts which are ones-over-many. Perhaps all one needs is the many, the plurality. But it seems one needs at least that to swerve as logical subject. If this is right, and to exist is to number more than zero, then we cannot sensibly say of an individual that it exists. We can say this of individuals but not of an individual. But surely we can say of an individual horse that it exists. So I conclude that 'exist(s)' cannot be a first-level non-distributive predicate.

3. And so one is driven to the conclusion that 'exist(s)' is a first-level distributive predicate. 'Horses exist' says of each individual horse that it exists. But isn't this equally objectionable? The vast majority of horses are such that I have no acquaintance with them at all. So how can my use of 'Horses exist' be about each horse?

It is at this juncture that Frege gets his revenge:

We must not think that I mean to assert something of an African chieftain from darkest Africa who is wholly unknown to me, when I say 'All men are mortal.' I am not saying anything about either this man or that man, but I am subordinating the concept man to the concept of what is mortal. In the sentence 'Plato is mortal' we have an instance of subsumption, in the sentence 'All men are mortal' one of subordination. What is being spoken about here is a concept, not an individual thing. (Posthumous Writings, p. 213)

Plato falls under the concept man; he does not fall within it. The concept mortal does not fall under the concept man — no concept is a man — but falls within it. When I say that all men are mortal I am not talking about individual men, but about the concept man, and I am saying that this concept has as part of its content the subconcept mortal.

Similarly, my utterance of 'Horses exist' cannot be about each horse; it is about the concept horse, and says that it has instances — which is the view I began by rejecting and for god reason.

We seem to have painted ourselves into an aporetic corner. No exit. Kein Ausgang. A-poria.

More on Translating ‘Something Exists’ and a Response to Brightly

I issued the following challenge: translate 'Something exists' into standard first-order predicate logic with identity. This is the logic whose sources are Frege and Russell. So I call it Frege-Russell logic, or, to be cute, 'Fressellian' logic. My esteemed commenters don''t see much of a problem here. So let me first try to explain why I see a problem. I then consider David Brightly's proposal.

1. First of all, 'Something exists' cannot be rendered as 'For some x, x exists.' This is because 'exist(s)' is not an admissible first-level predicate in Frege-Russell logic. The whole point of the Fressellian approach is to make 'exist(s)' disappear into the machinery of quantification. There is no such propositional function as 'x exists.' 'For some x, x exists' is gibberish, syntactic nonsense in Frege-Russell logic.

2. But the following is not gibberish: 'For some x, x = x.' So one will be tempted to say that 'Something exists' can be rendered as 'For some x, x = x,' ('Something is self-identical') and 'Everything exists' as 'For all x, x = x' ('Everything is self-identical').

But this won't work either. It is true that everything that exists is self-identical, and vice versa. But it doesn't follow, nor is it true, that existence is self-identity. Here is one consideration. When I say of Tom that he exists, I am not saying that he is self-identical. Suppose I hear a false rumour to the effect that Tom is no more. But then I encounter him in the flesh. I exclaim, "You still exist!" Clearly, "You are still self-identical" does not mean the same. If I said that, Tom might retort, "What the hell, man, were you worried that I had become legion?" In some circumstances, that a man should continue in existence is surprising. But we are never surprised by a man's continuing in self-identity.

Furthermore, when Tom ceases to exist, he does not become self-diverse. Loss of existence is not loss of self-identity. To put the point in formal mode, after his demise 'Tom' continues to refer to one and the same individual, Tom. The bearer of the name is gone, but not the reference. Otherwise it could not be true that Tom is gone. There is also a modal consideration. Tom is a contingent being: he exists but he might not have existed. If existence is self-identity, then Tom's possible nonexistence is Tom's possible self-diversity — which is absurd. It makes prima facie sense to say of an individual that it might not have existed or that it no longer exists; but it make no sense at all to say of an individual that it might not have been self-identical or that it is no longer self-identical. If Tom might not have existed, then it is Tom who might not have existed. But if Tom might not have been self-identical, then it is not Tom who might not have been self-identical.

So, even if everything that exists is self-identical and conversely, existence is not self-identity. When we say that something exists we are not saying that something is self-identical, and when we say that everything exists we are not saying that everything is self-identical. I conclude that 'Something exists' is not expressible in the terms of the Frege-Russell system. As for 'Everything exists,' it is surely a presupposition of the whole Frege-Russell approach: the approach presupposes that Meinong was wrong to speak of nonexistent objects. But this presupposition cannot be expressed, cannot be 'said,' in Fressellian terms.

We are in the following curious predicament. Something that must be true if if the Fresselian system is to be tenable — that everything exists, that there are no nonexistent objects — is not expressible within the system.

3. David Brightly accepts my challenge to give a Frege-Russell translation of 'Something exists.' He writes:

And as a Fressellian I accept the challenge. That property is Individual aka Object, the concept at the root of the Porphyrean tree. We can say 'Something exists' with ∃x.Object(x), ie, there is at least one object. Likewise ∀x.Object(x) (which is always true, even when the box is empty) says 'Everything exists' and its negation (which is always false) says 'Some thing is not an object'. But both these last are unenlightening—because always true and always false, respectively, they convey no information, make no distinction, are powerless to change us.

I asked: which property is it whose instantiation is the existence of something? David's answer is that it is the property or concept Individual or Object. And so I take David to be saying something like the following. "Just as the existence of cats is the being-instantiated of the concept cat, the existence of something is the being-instantiated of the concept Object."

David mentions the tree of Porphyry:

David speaks of the 'root' of the tree where I speak of its apex. No matter. However we visualize it, upside down or right side up, David's suggestion is that Object or Substance (as above) is a summum genus, a supreme genus. It is a concept superordinate to every concept, a concept under which everything falls.

Operating with a scheme like this, we can, in the spirit of Frege's dialogue with the illustrious Puenjer, reduce every existential proposition (or at least every general existential proposition) to a predication by climbing Porphyry's tree. Thus:

Cats exist –> Some mammal is a cat
Mammals exist –> Some animal is a mammal
Animals exist –> Some living thing is an animal
Living things exist –> Some body is a living thing
Bodies exist –> Some substance is a body
Substances exist –> Some Objects are substances.

The point of these translations is to dispense with 'existst(s)' by showing how propositions of the form Fs exist can be replaced salva veritate with propositions of the form Some G is a F, where G is superordinate to F. This amounts to the elimination of existence in favor of the logical quantity, someness.

We have now climbed to the tippy-top of the tree of Porphyry. We have ascended to a concept superordinate to every concept (except itself) a genus generalissimum, a most general genus. And what concept might that be? Such a concept must have maximal extension and so will have minimal intension. It will be devoid of all content, abstracting as it does from all differences. Frege in his dialog with Puenjer suggests something identical with itself as the maximally superordinate concept. 'There are men' and 'Men exist' thus get rendered as 'Something identical with itself is a man.' (63) Something identical with itself is equivalent to Brightly's Object.

4. Now why can't I accept the Frege-Brightly view? Well, I've already shown that 'Everything exists' cannot be translated as 'Everything is self-identical.' But this is tantamount to having established that the concept whose instantiation is the existence of everything cannot be the concept self-identical something or the concept Object.

Another way to see this is by considering two individuals at the very bottom of the Porphyrean tree. So consider my cats, Max and Manny. In respect of being cats, mammals, beasts, animals, living things, material substances, and self-identical somethings, they do not differ. They do not differ quidditatively. But they do differ: they differ in their very existence. Each has his own existence. Max is not Manny, and Manny is not Max. That is not a mere numerical difference; it is a numerical-existential difference. Since each cat has its own existence, the existence of either cannot be the being-instantiated of any quidditative concept. All such concepts abstract from existence. The same goes for all individuals. Individuals exist. But the existence of individuals is not the being-instantiated of any concept. If you want, you can think of existent (self-identical something) as a highest genus, but Existence — that in virtue of which things exist and are not nothing — is not a highest genus. And it is Existence that is the topic. There are no instances of Existence. Existing things are not a kind of thing.

The Frege-Russell theory fails utterly as a theory of Existence.

As sure as I am sitting here, I am sure that I will not convince the Londonistas. That fact is more grist for the (meta)philosophical mill.

On Translating ‘Some Individual Exists’ Fressellianly

An astute reader comments:

You write:

2. But can this presupposition be expressed (said) in this logic? Here is a little challenge for you Fressellians: translate 'Something exists' into standard logical notion. You will discover that it cannot be done. Briefly, if existence is instantiation, which property is it whose instantiation is the existence of something? Same problem with 'Nothing exists.' If existence is instantiation, which property is it whose non-instantiation is the nonexistence of anything? Similarly with 'Everthing exists' and 'Something does not exist.'

But couldn't we translate those expressions this way (assuming we have only two properties: a, b)?
1. "something exists" -> "there is an x that instantiates either a or b or ab"
2. "everything exists" -> "there is an x that instantiates a and there is a y that instantiates b and there is a z that instantiates ab"
3. "nothing exists" -> 1 is false
4. "something doesn't exist" -> 2 is false

I am afraid that doesn't work. We need focus only on on 'Some individual exists.' The reader's proposal could be put as follows. Given the properties F-ness and G-ness,

What 'Some individual exists' says is exactly what 'Either F-ness is instantiated or G-ness is instantiated' says.

I would insist however that they do not say the same thing, i.e., do not have the same meaning. The expression on the left says that some individual or other, nature unspecified, exists. The expression on the right, however, makes specific reference to the 'natures' F-ness and G-ness. Surely, 'Some individual exists' could be true even if there are are no individuals that are either Fs or Gs.

Note that it is not a matter of logic what properties there are. This is an extralogical question.

On the Frege-Russell treatment of existence, 'exist(s)' is a second-level predicate, a predicate of concepts, properties, propositional functions and cognate items. It is never an admissible predicate of individuals. Thus in this logic every affirmation of existence must say of some specified concept or property that it is instantiated, and every denial of existence must say of some specified concept or property that it fails of instantiation.

This approach runs into trouble when it comes to the perfectly meaningful and true 'Something exists' and 'Some individual exists.' For in these instances no concept or property can be specified whose instantiation is the existence of things or the existence of individuals. To head off an objection: self-identity won't work.

That there are individuals is a necessary presupposition of the Frege-Russell logic in that without it one cannot validly move from 'F-ness is instantiated' to 'Fs exist.' But it is a necessary presupposition that cannot be stated in the terms of the system. This fact, I believe, is one of the motivations for Wittgenstein's distinction between the sayable and the showable. What cannot be said, e.g., that there are individuals, is shown by the use of such individual variables as 'x.'

The paradox, I take it, is obvious. One cannot say that 'There are individuals' is inexpressible without saying 'There are individuals.' When Wittgenstein assures us that there is the Inexpressible, das Unaussprechliche, he leaves himself open to the retort: What is inexpressible? If he replies, 'That there are individuals,' then he is hoist by his own petard.

Surely it is true that there are individuals and therefore expressible, because just now expressed.

"The suicide of a thesis," says Peter Geach (Logic Matters, p. 265), "might be called Ludwig's self-mate . . . . " Here we may have an instance of it.

Transitivity of Predication?

I dedicate this post to London Ed, who likes sophisms and scholastic arcana.

Consider these two syllogistic arguments:

A1. Man is an animal; Socrates is a man; ergo, Socrates is an animal.
A2. Man is a species; Socrates is a man; ergo, Socrates is a species.

The first argument is valid. On one way of accounting for its validity, we make two assumptions. First, we assume that each of the argument's constituent sentences is a predication. Second, we assume the principle of the Transitivity of Predication: if x is predicable of y, and y is predicable of z, then x is predicable of z. This principle has an Aristotelian pedigree. At Categories 3b5, we read, "For all that is predicated of the predicate will be predicated also of the subject." So if animal is predicable of man, and man of Socrates, then animal of Socrates.

Something goes wrong, however, in the second argument. The question is: what exactly? Let's first of all see if we can diagnose the fallacy while adhering to our two assumptions. Thus we assume that each occurrence of 'is' in (A2) is an 'is' of predication, and that predication is transitive. One suggestion — and I take this to be the line of some Thomists — is that (A2) equivocates on 'man.' In the major, 'man' means 'man-in-the-mind,' 'man as existing with esse intentionale.' In the minor, 'man' means 'man-in-reality,' 'man as existing with esse naturale.' We thus diagnose the invalidity of (A2) by saying that it falls afoul of quaternio terminorum, the four-term fallacy. On this diagnosis, Transitivity of Predication is upheld: it is just that in this case the principle does not apply since there are four terms.

But of course there is also the modern Fregean way on which we abandon both of our assumptions and locate the equivocation in (A2) elsewhere. On a Fregean diagnosis, there is an equivocation on 'is' in (A2) as between the 'is' of inclusion and the 'is' of predication. In the major premise, 'is' expresses, not predication, but inclusion: the thought is that the concept man includes within its conceptual content the subconcept species. In the minor and in the conclusion, however, the 'is' expresses predication: the thought is that Socrates falls under the concepts man and species. Accordingly, (A2) is invalid because of an equivocation on 'is,' not because of an equivocation on 'man.'

The Fregean point is that the concept man falls WITHIN but not UNDER the concept animal, while the object Socrates falls UNDER but not WITHIN the concepts man and animal. Man does not fall under animal because no concept is an animal. Animal is a mark (Merkmal) not a property (Eigenschaft) of man. In general, the marks of a concept are not its properties. But concepts do have properties. The property of being instantiated, for example, is a property of the concept man. But it is not a mark of it. If it were a mark, then man by its very nature would be instantiated and it would be a conceptual truth that there are human beings, which is false.

Since on the Fregean scheme the properties of concepts needn't be properties of the items that fall under the concepts, Transitivity of Predication fails. Thus, the property of being instantiated is predicable of the concept philosopher, and the concept philosopher is predicable of Socrates; but the property of being instantiated is not predicable of Socrates.

Frege Meets Aquinas: A Passage from De Ente et Essentia

Here is a passage from Chapter 3 of Thomas Aquinas, On Being and Essence (tr. Robert T. Miller, emphasis added):

The nature, however, or the essence thus understood can be considered in two ways. First, we can consider it according to its proper notion, and this is to consider it absolutely. In this way, nothing is true of the essence except what pertains to it absolutely: thus everything else that may be attributed to it will be attributed falsely. For example, to man, in that which he is a man, pertains animal and rational and the other things that fall in his definition; white or black or whatever else of this kind that is not in the notion of humanity does not pertain to man in that which he is a man. Hence, if it is asked whether this nature, considered in this way, can be said to be one or many, we should concede neither alternative, for both are beyond the concept of humanity, and either may befall the conception of man. If plurality were in the concept of this nature, it could never be one, but nevertheless it is one as it exists in Socrates. Similarly, if unity were in the notion of this nature, then it would be one and the same in Socrates and Plato, and it could not be made many in the many individuals. Second, we can also consider the existence the essence has in this thing or in that: in this way something can be predicated of the essence accidentally by reason of what the essence is in, as when we say that man is white because Socrates is white, although this does not pertain to man in that which he is a man.

What intrigues me about this passage is the following argument that it contains:

1. A nature can be considered absolutely (in the abstract) or according to the being it has in this or that individual.
2. If a nature is considered absolutely, then it is not one. For if oneness were included in the nature of humanity, e.g., then humanity could not exist in many human beings.
3. If a nature is considered absolutely, then it is not many. For if manyness were included in the nature of humanity, e.g., then humanity could not exist in one man, say, Socrates.
Therefore
4. If a nature is considered absolutely, then it is neither one nor many, neither singular nor plural.

The following is clear. There are many instances of humanity, many human beings. Therefore, there can be many such instances. It follows that there is nothing in the nature of humanity to preclude there being many such instances. But there is also nothing in the nature of humanity to require that there be many instances of humanity, or even one instance. We can express this by saying that the nature humanity neither requires nor precludes its being instantiated. This nature, considered absolutely, logically allows multiple instantiation, single instantiation, and no instantiation. It logically allows that there be many men, just one man, or no men.

But surely it does not follow that the nature humanity is neither one nor many. What Aquinas is doing above is confusing what Frege calls a mark (Merkmal) of a concept with a property (Eigenschaft) of a concept. The marks of a concept are the subconcepts which are included within it. Thus man has animal and rational as marks. But these are not properties of the concept man since no concept is an animal or is rational. Being instantiated is an example of a property of man, a property that cannot be a mark of man. In general, the marks of a concept are not properties thereof, and vice versa. Exercise for the reader: find a counterexample, a concept which is such that one of its marks is also a property of it.

Aquinas has an insight which can be expressed in Fregean jargon as follows. Being singly instantiated — one in reality — and being multiply instantiated — many in reality — are not marks (Merkmale) of the nature humanity. But because he (along with everyone else prior to 1884) confuses marks with properties (Eigenschaften), he concludes that the nature itself cannot be either one or many.

For these reasons I do not find the argument from De Ente et Essentia compelling. But perhaps there is a good Thomist response.

Kenny, Geach, and the Perils of Reading Frege Back Into Aquinas

Frege, like Aquinas, held that there was a fundamental distinction in rebus answering to the logical distinction between subject and predicate — the distinction between Gegenstand (object) and Begriff (concept). [. . .] And for Frege the Begriff, and it alone, admits of repetition and manyness; an object cannot be repeated — kommt nie wiederholdt vor. (45-46)

So far, so good. Geach continues:

Understood in this way, the distinction between individual and form is absolutely sharp and rigid; what can be sensibly said of one becomes nonsense if we try to say it of the other. [. . .] Just because of this sharp distinction, we must reject the Platonic doctrine that what a predicate stands for is is some single entity over against its many instances, hen epi pollon. On the contrary: the common nature that the predicate 'man' (say) stands for can be indifferently one or many, and neither oneness nor manyness is a mark or note of human nature itself. This point is made very clearly by Aquinas in De Ente et Essentia. Again we find Frege echoing Aquinas; Frege counts oneness or manyness (as the case may be) among the properties (Eigenschaften) of a concept, which means that it cannot at the same time be one of the marks or notes (Merkmalen) of that concept. (46)

Morning Star and Evening Star

London Ed of Beyond Necessity does a good job patiently explaining the 'morning star' – 'evening star' example to one of his uncomprehending readers. But I don't think Ed gets it exactly right. I quibble with the following:

Summarising:
(1) The sentence “the morning star is the evening star” has informational content.
(2) The sentence “the morning star is the morning star” does not have informational content.
(3) Therefore, the term “the morning star” does not have the same informational content as “the evening star”.

One quibble is this. Granted, the two sentences differ in cognitive value, Erkenntniswert. (See "On Sense and Reference" first paragraph.) The one sentence expresses a truth of logic, and thus a truth knowable a priori. The other sentence expresses a factual truth of astronomy, one knowable only a posteriori. But note that Frege says that they differ in cognitive value, not that the one has it while the other doesn't. Ed says that the one has it while the other doesn't — assuming Ed is using 'informational content' to translate Erkenntniswert. There is some annoying slippage here.

More importantly, I don't see how cognitive value/informational content can be had by such subsentential items as 'morning star' and 'evening star.' Thus I question the validity of the inference from (1) & (2) to (3). Neither term gives us any information. So it cannot be that they differ in the information they give. Nor can they be contrasted in point of giving or not giving information. Information is conveyable only by sentences or propositions.

I say this: neither of the names Morgenstern (Phosphorus) or Abendstern (Hesperus) have cognitive value or informational content. (The same holds, I think, if they are not proper names but definite descriptions.) Only indicative sentences (Saetze) and the propositions (Gedanken) they express have such value or content. As I see it, for Frege, names have sense (Sinn) and reference (Bedeutung), and they may conjure up subjective ideas (Vorstellungen) in the minds of their users. But no name has cognitive value. Sentences and propositions, however, have sense, reference, and cognitive value. Interestingly, concept-words (Begriffswoerter) or predicates also have sense and reference, but no cognitive value.

I also think Ed misrepresents the Compositionality Principle. Frege is committed to compositionality of sense (Sinn), not compositionality of informational content/cognitive value. So adding the C. P. to his premise set will not validate the above inference.

The Difference Between a Truth-Bearer and a Truth-Maker

Frege makes the point that the being of a proposition cannot be identical to its being true. This I find obvious. There are true propositions and there are false propositions. Therefore, for propositions (the senses of context-free declarative sentences) it cannot be the case that to be = to be true. Furthermore, a given proposition that is contingently true is possibly such as not to be true, whence it follows that its being and its being true cannot be identical. (Whether Frege does or would give the second argument, I don't know; but I think it is correct.)

As Frege puts its, "The being of a thought [Gedanke, proposition] thus does not consist in its being true." (Near the beginning of his essay, "Negation.") One can grasp a proposition without knowing whether or not it is true. To grasp a proposition is not to accept it as true, to reject it as false, or to suspend judgment as to its truth-value. To grasp a proposition is merely to have it before one's mind, to understand it. A Fregean proposition is a sense, and no such propositional sense has as part of its sense its being true. That's Frege's point and it strikes me as rock-solid.

Our London sparring-partner Ed now demonstrates that he still does not understand what a truth-maker is:

I wonder if a ‘truthmaker’ as understood by the advocates of truthmaking is the same sort of thing as Frege’s marvelous but impossible thought. Something that if we perceived it for what it was, would simultaneously communicate to us the truth of what it includes.

Ed is obviously confusing truth-bearers such as Fregean propositions with truth-makers. Truth-bearers are representations; truth-makers are not. That's one difference. Truth-bearers are either true or false; truth-makers are not since, not being representations, they cannot be said to be true, nor can they be said to be false. That's a second difference. Truth-bearers are 'bipolar,' either true or false; truth-makers are 'unipolar': all of them obtain. That's a third difference. Truth-bearers are such that their being or existence does not entail their being true; truth-makers are such that their being or existence does entail their obtaining. I am assuming that truth-makers are facts. If a fact obtains then it exists; there are no non-obtaining facts. That's a fourth difference.

There is no point in criticizing a doctrine one misrepresents. First represent it fairly, then lodge objections. And as I have said, there are reasonable objections one can bring.

So far Ed has not lodged one clear objection.

Frege’s Regress

Some of us of a realist persuasion hold that at least some truths have need of worldly correlates that 'make them true.' This notion that (some) truths need truthmakers is a variation on the ancient theme that truth implies a correspondence of what-is said or what-is-thought with what-is. You all know the passages in Aristotle where this theme is sounded.

Example. Having just finished my drink, the thought expressed by an assertive utterance of 'My glass is empty' is true. But the thought is not just true; it is true because of the way things are 'outside' my mind. The glass (in reality) is (in reality) empty. So the realist says something like this: the thought (proposition, judgmental content, etc.) is true in virtue of the obtaining of a truthmaking state of affairs or fact. The thought is true because the fact obtains or exists, where 'because' does not have a causal sense but expresses the asymmetrical relation of truthmaking. The fact is the ontological ground (not the cause) of the thought's being true.

One might wonder whether this realist theory of truth leads to an infinite regress, and if it does, whether the regress is vicious. Some cryptic remarks in Gottlob Frege's seminal article, "The Thought: A Logical Inquiry," suggest a regress argument against the correspondence theory of truth.

For Frege, a thought (Gedanke) or proposition is the sense (Sinn) of a context-free declarative sentence. 'Snow is white' and its German translation Schnee ist weiss are examples of context-free declarative sentences. 'Context-free' means that all indexical elements have been extruded including verb tenses. When we say that a sentence such as 'Snow is white' is true, what we are really saying is that the sense of this sentence is true. The primary truth-vehicles are propositions, sentences being truth-bearers only insofar as they express true propositions.

Now could the being-true of a sentential sense consist in its correspondence to something else? Frege rejects this notion: "In any case, being true does not consist in the correspondence of this sense with something else, for otherwise the question of truth would reiterate itself to infinity." (Philosophical Logic, ed. Strawson, p. 19) A little earlier, Frege writes,

For what would we then have to do to decide whether something were true? We should have to enquire whether it were true that an idea and a reality, perhaps, corresponded in the laid-down respect. And then we should be confronted by a question of the same kind and the game could begin again. So the attempt to explain truth as correspondence collapses. And every other attempt to define truth collapses too. (Ibid.)

What exactly is Frege's argument here? We begin by noting that

1. Necessarily, for any proposition p, it is true that p iff p.

This equivalence, which I hope nobody will deny, gives rise to an infinite regress, call it the truth regress. For from (1) we can infer that if snow is white, then it is true that snow is white, and
iterating the operation, if it is true that snow is white, then it is true that it is true that snow is white, and so on without end. This is an infinite regress all right, but it is obviously benign. For if
we establish the base proposition, Snow is white, then we ipso facto establish all the iterations. Our establishing that snow is white does not depend on a prior establishing that it is true that snow is white. In general, our establishing of any proposition in the infinite series does not depend on having first established the next proposition in the series. The truth regress, though infinite, is benign.

Note that if the truth-regress were vicious, then the notion of truth itself would have been shown to be incoherent. For the truth-regress is a logical consequence of the equivalence principle (1) above, a principle that simply unpacks our understanding of 'true.' So if the truth-regress were vicious, then (1) would not be unproblematic, as it surely is.

It follows that if Frege's Regress is to amount to a valid objection to the definition of truth as correspondence, "and [to] every other attempt to define truth," then Frege's Regress must be different from the truth regress. In particular, it must be a vicious regress. Only vicious infinite regresses have the force of philosophical refutations. But then what is Frege's Regress? Consider

2. Necessarily, for any p, it is true that p iff *p* corresponds to reality.

One can think up counterexamples to (2), but the precise question before us is whether (2) issues in a vicious infinite regress. Now what would this regress (progress?) look like? Let 'T(p)' abbreviate
'it is true that p.' And let 'C*p*' abbreviate '*p* corresponds to reality.' (The asterisks function like Quine's corners.) The regress, then, looks like this:

3. p iff T(p) iff C*p* iff T(C*p*) iff C(T(C*p*)) iff T(C(T(C*p*) iff C(T(C(T(C*p*)) . . .

Is (3) a vicious regress? It would be vicious if one could establish T(p) only by first establishing C*p* and so on. But if these two terms have the same sense, in the way that the first and second terms have the same sense, then (3) will be as benign as the truth regress. Suppose that 'It is true that p' and '*p* corresponds to reality' have the same sense. Suppose in other words that the correspondence theory of truth is the theory that the sense or meaning of these distinct sentences is the same. It would then follow that to establish that it is true that p and to establish that *p* corresponds to reality would come to the same thing, whence it would follow that the regress is benign.

For the regress to be vicious, the second and third terms must differ in sense. For again, if the second and third terms do not differ in sense, then to establish one is to establish the other, and it would not be case that to establish that it is true that p one would first have to establish that *p* corresponds to reality or to some chunk of reality. But if the second and third terms do not differ in sense, then it appears that the regress doesn't get started at all. For the move from the second term to the third to be valid, the entailment must be grounded in the sense of the second term: the third term must merely unpack the sense of the second term. If, however, the two terms are not sense-connected, then no infinite regress is ignited.

My interim conclusion is that it is not at all clear that Frege's Regress is either benign, or not a regress at all, and therefore not at all clear that it constitutes a valid objection to theories of truth, in particular to the theory that truth resides in correspondence.

REFERENCE: Peter Carruthers, "Frege's Regress," Proc. Arist. Soc., vol. LXXXII, 1981/1982, pp. 17-32.

Atomic Sentences and Syncategorematic Elements

According to Fred Sommers (The Logic of Natural Language, p. 166), ". . . one way of saying what an atomic sentence is is to say that it is the kind of sentence that contains only categorematic expressions." Earlier in the same book, Sommers says this:

In Frege, the distinction between subjects and predicates is not due to any difference of syncategorematic elements since the basic subject-predicate propositions are devoid of such elements. In Frege, the difference between subject and predicate is a primitive difference between two kinds of categorematic expressions. (p. 17)

Examples of categorematic (non-logical) expressions are 'Socrates' and 'mammal.' Examples of syncategorematic (logical) expressions are 'not,' 'every,' and 'and.' As 'syn' suggests, the latter expressions are not semantic stand-alones, but have their meaning only together with categorematic expressions. Sommers puts it this way: "Categorematic expressions apply to things and states of affairs; syncategorematic expressions do not." (164)

At first I found it perfectly obvious that atomic sentences have only categorematic elements, but now I have doubts. Consider the atomic sentence 'Al is fat.' It is symbolized thusly: Fa. 'F' is a predicate expression the reference (Bedeutung) of which is a Fregean concept (Begriff) while 'a' is a subject-expression or name the reference of which is a Fregean object (Gegenstand). Both expressions are categorematic or 'non-logical.' Neither is syncategorematic. And there are supposed to be no syncategorematic elements in the sentence: there is just 'F' and 'a.'

But wait a minute! What about the immediate juxtaposition of 'F' and 'a' in that order? That juxtaposition is not nothing. It conveys something. It conveys that the referent of 'a' falls under the referent of 'F'. It conveys that the object a instantiates the concept F. I suggest that the juxtaposition of the two signs is a syncategorematic element. If this is right, then it is false that atomic sentence lack all syncategorematic elements.

Of course, there is no special sign for the immediate juxtaposition of 'F' and 'a' in 'Fa.' So I grant that there is no syncategorematic element if such an element must have its own separate and isolable sign. But there is no need for a separate sign; the immediate juxtaposition does the trick. The syncategorematic element is precisely the juxtaposition.

Please note that if there were no syncategorematic element in 'Fa' there would not be any sentence at all. A sentence is not a list. The sentence 'Fa' is not the list 'F, a.' A (declarative) sentence expresses a thought (Gedanke) which is its sense (Sinn). And its has a reference (Bedeutung), namely a truth value (Wahrheitswert). No list of words (or of anything else) expresses a thought or has a truth value. So a sentence is not a list of its constituent words. A sentence depends on its constituent words, but it is more than them. It is their unity.

So I say there must be a syncategorematic element in 'Fa' if it is to be a sentence. There is need of a copulative element to tie together subject and predicate. It follows that, pace Sommers, it is false that atomic sentences are devoid of syntagorematic elements.

Note what I am NOT saying. I am not saying that the copulative element in a sentence must be a separate sign such as 'is.' There is no need for the copulative 'is.' In standard English we say 'The sea is blue' not 'The sea blue.' But in Turkish one can say Deniz mavi and it is correct and intelligible. My point is not that we need the copulative 'is' as a separate sign but that we need a copulative element which, though it does not refer to anything, yet ties together subject and predicate. There must be some feature of the atomic sentence that functions as the copulative element, if not immediate juxtaposition then something else such as a font difference or color difference.

At his point I will be reminded that Frege's concepts (Begriffe) are unsaturated (ungesaettigt). They are 'gappy' or incomplete unlike objects. The incompleteness of concepts is reflected in the incompleteness of predicate expressions. Thus '. . . is fat' has a gap in it, a gap fit to accept a name such as 'Al' which has no gap. We can thus say that for Frege the copula is imported into the predicate. It might be thought that the gappiness of concepts and predicate expressions obviates the need for a copulative element in the sentence and in the corresponding Thought (Gedanke) or proposition.

But this would be a mistake. For even if predicate expressions and concepts are unsaturated, there is still a difference between a list and a sentence. The unsaturatedness of a concept merely means that it combines with an object without the need of a tertium quid. (If there were a third thing, then Bradley's regress would be up and running.) But to express that a concept is in fact instantiated by an object requires more than a listing of a concept-word (Begriffswort) and a name. There is need of a syncategorical element in the sentence.

So I conclude that if there are any atomic sentences, then they cannot contain only categorematic expressions.

Nota Notae Est Nota Rei Ipsius and the Ontological Argument

(By popular demand, I repost the following old Powerblogs entry.)

"The mark of a mark is a mark of the thing itself." I found this piece of scholasticism in C. S. Peirce. (Justus Buchler, ed., Philosophical Writings of Peirce, p. 133) It is an example of what Peirce calls a 'leading principle.'

Let's say you have an enthymeme:

   Enoch was a man
   —–
   Enoch died.

Invalid as it stands, this argument can be made valid by adding a premise. (Any invalid argument can be made valid by adding a premise.) Add 'All men die' and the argument comes out valid. Peirce writes:

The leading principle of this is nota notae est nota rei ipsius.
Stating this as a premiss, we have the argument,

     Nota notae est nota rei ipsius
     Mortality is a mark of humanity, which is a mark of Enoch
     —–
     Mortality is a mark of Enoch.

But is it true that the mark of a mark is a mark of the thing itself? There is no doubt that mortality is a mark of humanity in the following sense: The concept humanity includes within its conceptual content the superordinate concept mortal, which implies that, necessarily, if anything is human, then it is mortal. But mortality is not a mark, but a property, of Enoch. I am alluding to Frege's distinction between a Merkmal and an Eigenschaft. Frege explains this distinction in various places, one being The Foundations of Arithmetic, sec. 53. But rather than quote Frege, I'll explain the distinction in my own way using a totally original example.

Consider the concept bachelor. This is a first-order or first-level concept in that the items that fall under it are not concepts but objects. The marks of a first-order concept are properties of the objects that fall under the concept. Now the marks of bachelor are unmarried, male, adult, and not a member of a religious order. These marks are themselves concepts, concepts one can extract from bachelor by analysis. Given that Tom falls under bachelor, he has these marks as properties. Thus unmarried, etc. are not marks of Tom, but properties of Tom, while unmarried, etc. are not properties of bachelor but marks of bachelor.

To appreciate the Merkmal (mark)-Eigenschaft (property) distinction, note that the relation between a concept and its marks is entirely different from the relation between a concept and its instances. A first-order concept includes its marks without instantiating them, while an object instantiates its properties without including them.

This is a very plausible line to take. It makes no sense to say of a concept that it is married or unmarried, so unmarried cannot be a property of the concept bachelor. Concepts don't get married or remain single. But it does make sense to say that a concept includes certain other concepts, its marks. On the other hand, it makes no sense to say of Tom that he includes certain concepts since he could do such a thing only if he were a concept, which he isn't. But it does make sense to say of Tom that he has such properties as being a bachelor, being unmarried, being an adult, etc.

Reverting to Peirce's example, mortality is a mark of humanity, but not a mark of Enoch. It is a property of Enoch. For this reason the scholastic formula is false. Nota notae NON est nota rei ipsius. The mark of a mark is not a mark of the thing itself but a property of the thing itself.

No doubt commenter Edward the Nominalist will want to wrangle with me over this slight to his scholastic lore, and I hope he does, since his objections will aid and abet our descent into the labyrinth of this fascinating cluster of problems. But for now, two quick applications.

One is to the ontological argument, or rather to the ontological argument aus lauter Begriffen as Kant describes it, the ontological argument "from mere concepts." So we start with the concept of God and analyze it. God is omniscient, etc. But 'surely' existence is also contained in the concept of God. For a God who did not exist would lack a perfection, a great-making property; such a God would not be id quo maius cogitari non posse. He would not be that than which no greater can be conceived. To conceive God, then, is to conceive an existing God, whence it follows that God exists! For if you are conceiving a nonexistent God, then you are not conceiving God.

Frege refutes this version of the OA — not the only or best version I hasten to add — in one sentence: Weil Existenz Eigenschaft des Begriffes ist, erreicht der ontologische Beweis von der Existenz Gottes sein Ziel nicht. (Grundlagen der Arithmetik, sec. 53) "Because existence is a property of concepts, the ontological argument for the existence of God fails to attain its goal." What Frege is saying is that the OA "from mere concepts" rests on the mistake of thinking of existence as a mark of concepts as opposed to a property of concepts. No concept for Frege is such that existence is included within it. Existence is rather a property of concepts, the property of having an instance.

The other application of my rejection of the scholastic formula above is to the logical question of the correct interpretation of singular propositions. The scholastics treat singulars as if they are generals as I explained fully in previous posts. But if Frege is right, this is a grave logical error since it rides roughshod over the mark/property distinction. To drag this all into the full light of day will take many more posts.